MELLIN-BARNES INTEGRALS FOR STABLE DISTRIBUTIONS AND THEIR CONVOLUTIONS Francesco Mainardi 1 and Gianni Pagnini 2 Dedicated to Professor Anatoly A. Kilbas Dept. of Mathematics and Mechanics, University of Minsk, Belarus, on the occasion of his 60th birthday (20 July 2008) Abstract In this expository paper we ¯rst survey the method of Mellin-Barnes integrals to represent the ® stable L¶evydistributions in probability theory (0 < ® · 2). These integrals are known to be useful for obtaining conver- gent and asymptotic series representations of the corresponding probability density functions. The novelty concerns the convolution between two stable probability densities of di®erent L¶evyindex, which turns to be a probabil- ity law of physical interest, even if it is no longer stable and self-similar. A particular but interesting case of convolution is obtained combining the Cauchy-Lorentz density (® = 1) with the Gaussian density (® = 2) that yields the so-called Voigt pro¯le. Our machinery can be applied to derive the fundamental solutions of space-fractional di®usion equations of two or- ders. 2000 Mathematics Subject Classi¯cation: 26A33, 33C60, 42A38, 44A15, 44A35, 60G18, 60G52 Key Words and Phrases: Mellin-Barnes integrals, Mellin transforms, stable distributions, Voigt pro¯le, space-fractional di®usion 444 F. Mainardi, G. Pagnini Introduction Mellin-Barnes integrals are a family of integrals in the complex plane whose integrand is given by the ratio of products of Gamma functions. De- spite of the name, the Mellin-Barnes integrals were initially introduced in 1888 by the Italian mathematician S. Pincherle [27] in a couple of papers on the duality principle between linear di®erential equations and linear dif- ference equations with rational coe±cients, as discussed by the Authors in [20]. Only at the beginning of the XX century they were widely adopted by Mellin and Barnes. The Mellin-Barnes integrals are strongly related with the Mellin transform, in particular with its inverse transformation, in the framework of the so-called Melling setting. As shown in [22], [29], see also [10], [16], [32], [36], a powerful method to evaluate integrals can be obtained based on the Mellin setting. The Mellin-Barnes integrals are also essential tools for treating higher transcendental functions as generalized hyperge- ometric functions pFq and Meijer G- and Fox H-functions, see [17], [18], [23], [38]. Furthermore, they provide a useful representation to compute the asymptotic behaviour of functions, see [26], [40]. All the above machinery is also used in the theory of probability, see e.g. [21], [24], [34], [37], [39], [41], [42], and in theory of Fractional Calculus, see e.g. [4], [5], [6], [7], [8], [9], [15], [19], [28], [35]. We point out the forthcoming handbook on Mellin Transforms by Brychkov, Kilbas, Marichev and Prudnikov [3], that will surely become a table-text for most applied mathematicians. In this paper, after the essential notions and notations concerning the Mellin-Barnes integrals and the Mellin transform, we recall how the ® sta- ble distributions in probability theory (0 < ® · 2) can be represented by these tools. In the framework of symmetric distributions we illustrate our simple method to derive the known results. Then, we extend the method to derive the Mellin-Barnes representation for the probability den- sities obtained through a convolution of two stable densities of index ®1; ®2 (0 < ®1 < ®2 · 2). The case f®1 = 1 ; ®2 = 2g is of physical interest since it is related to the so-called Voigt pro¯le function, well-known in molecular spectroscopy and atmospheric radiative transfer. We show how the convolu- tion is of general interest since it enters in the treatment of space-fractional di®usion equations of double order. Finally, we point out that our results, being based on simple manipulations, can be understood by non-specialists of transform methods and special functions; however they could be derived through a more general analysis involving the functions of the Fox type. MELLIN-BARNES INTEGRALS FOR STABLE DISTRIBUTIONS ...445 1. The Mellin setting: basic formulas The Mellin-Barnes integrals are complex integrals which contain Gamma functions in their integrands as follows, Z γ+i1 1 ¡(a1 + A1s) ¢ ¢ ¢ ¡(am + Ams) MB§(z) = 2¼i γ¡i1 ¡(c1 + C1s) ¢ ¢ ¢ ¡(cp + Cps) (1:1) ¡(b ¡ B s) ¢ ¢ ¢ ¡(b ¡ B s) £ 1 1 n n z§s ds : ¡(d1 ¡ D1s) ¢ ¢ ¢ ¡(dq ¡ Dqs) It is assumed that γ is real, all the Aj, Bj, Cj, Dj are positive, and all the aj, bj, cj, j are complex. The path of integration is a straight line parallel to the imaginary axis with indentations, if necessary, to avoid the poles of the integrands. For more details see, e.g. [2], [11], [12], [26]. By using the residue theorem it is not di±cult to formally expand these integrals in power series. The Melin-Barnes integrals are known to occur in the de¯nitions of higher transcendental functions as generalized hypergeometric functions pFq and Meijer G- and Fox H-functions, see [17], [18], [23], [38], being related to their Mellin transforms. For convenience, let us here recall the essential formulas for this kind of integral transform, referring for details to specialized textbooks, e.g., [10], [16], [22], [3], [36]. Let Z +1 ¤ s¡1 M ff(x); sg = f (s) = f(x) x dx; γ1 < < (s) < γ2; (1:2) 0 be the Mellin transform of a su±ciently well-behaved function f(x) ; and let Z 1 γ+i1 M¡1 ff ¤(s); xg = f(x) = f ¤(s) x¡s ds (1:3) 2¼i γ¡i1 be the inverse Mellin transform, where x > 0 ; γ = < (s) ; γ1 < γ < γ2 : It is straightforward to note that MB¡(x) provide an essential tool for the inversion of Mellin transforms when these are expressed in terms products and ratios of Gamma functions. Denoting by $M the juxtaposition of a function f(x) with its Mellin transform f ¤(s) ; the main rules are: f(ax) $M a¡s f ¤(s) ; a > 0 ; (1:4) 446 F. Mainardi, G. Pagnini xa f(x) $M f ¤(s + a) ; (1:5) 1 f(xp) $M f ¤(s=p) ; p 6= 0 ; (1:6) jpj Z1 1 h (x) = f(y) g(x=y) dy $M h¤(s) = f ¤(s) g¤(s) ; (1:7) 1 y 1 0 Z1 M ¤ ¤ ¤ h2(x) = f(xy) g(y) dy $ h2(s) = f (s) g (1 ¡ s) : (1:8) 0 The most simple example of Mellin transform is provided by the Legendre integral representation of the Gamma function Z 1 ¡(s) = e¡x xs¡1 dx ; <(s) > 0 ; so e ¡x $M ¡(s) : (1:9) 0 Henceforth, the Mellin-Barnes integral representation of exp(¡z) turns to be Z 1 X1 (¡1)n e ¡z = ¡(s) z¡s ds = zn ; (1:10) 2¼i n! L n=0 where L denotes a loop in the complex s plane which encircles the poles of ¡(s) (in the positive sense) with endpoints at in¯nity in <(s) < 0 and with no restrictions on arg z. 2. Stable distributions and their Mellin setting The topic of stable distributions is a fascinating and fruitful area of research in probability theory; furthermore, nowadays, these distributions provide valuable models in physics, astronomy, ¯nance, and communication theory. The general class of stable distributions was introduced and given this name by the French mathematician Paul L¶evyin the early 1920's. Stable distributions have three \exclusive" properties, which can be briefly summarized stating that they 1) are \invariant under addition", 2) possess their own \domain of attraction", and 3) admit a "canonical characteristic function". Referring to specialized textbooks for more details, see e.g. [1], [13], [31], [33], [39], [42], here we limit ourselves to consider the class of L¶evy MELLIN-BARNES INTEGRALS FOR STABLE DISTRIBUTIONS ...447 strictly stable densities according to the Feller parameterization. Following Mainardi et al. [19], this class will be denoted by θ fL®(x)g ; with 0 < ® · 2 ; jθj · min f®; 2 ¡ ®g ; x 2 R ; (2:1) where ® denotes the index of stability (or L¶evyindex) and θ is a real pa- rameter related to the asymmetry, improperly referred to as the skewness. Then, the canonical characteristic function, namely the Fourier transform of the density (2.1), is Z +1 h i θ cθ i·x θ θ F fL®(x); ·g = L®(·) := e L®(x) dx = exp ¡Ã®(·) ; (2:2) ¡1 where θ ® i(sign ·)θ¼=2 θ ¡θ î(·) := j·j e = î(¡·) = î (¡·) ;· 2 R : (2:3) It is easy to note from their characteristic functions that the strictly stable densities are self-similar. Indeed, by setting with a > 0, h i θ F ® i(sign ·)θ¼=2 L®(x; a) $ exp ¡aj·j e ; (2:4) we have ³ ´ θ ¡1=® θ 1=® L®(x; a) = a L® x=a : (2:5) For θ = 0 we obtain symmetric densities of which noteworthy examples are provided by the Gaussian (or normal) law (with ® = 2) and the Cauchy- Lorentz law (® = 1). The corresponding expressions are usually given as 2 2 1 ¡x =(2σ ) 0 2 pG(x; σ) := p e := L (x; a = σ =2) ; x 2 R ; (2:6) 2¼ σ 2 where σ2 denotes the variance, and 1  p (x; Â) := := L0(x; a = Â) ; x 2 R ; (2:7) CL ¼ (x2 + Â2) 1 where  denotes the semi-interquartile range. Convergent and asymptotic series expansions for stable densities were introduced by Feller [13] in the 1950's without be classi¯ed in the frame- work of a known class of special functions. A general representation of all stable distributions was only achieved in 1986 by Schneider [34], who, in his 448 F.